Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

As we know, there's a structure theorem for modules over $\mathbb{Z}$. What happens over $\mathbb{Z}^2$, or more generally, over $\mathbb{Z}^n$? Obviously, these rings are not integral domains, but I wonder if there are some results concerning the structure of modules over them.

Thanks Alex

share|improve this question

1 Answer 1

Whenever you have two unital rings $R$ and $S$, then $e_R=(1,0)$ and $e_S=(0,1)$ are central idempotents in $R\times S$ and every module $M$ decomposes as $e_RM\oplus e_SM$. Now $e_RM$ is an $R$-module and $e_SM$ is an $S$-module.

Then in your case you can apply the structure theorem.

For more detailed proofs, see e.g. Ringel, Schröer, Proposition 12.16.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.